The strength of a local gravitational field depends upon the proximity to the mass of an object(s). The mass of an object, in turn, is dependent upon the density of the material of the object and the volume of the object. Accordingly, density variations of geological structures such as mineral deposits, oil reservoirs, underground tunnels or caverns have specific gravity signatures. These signatures, if measured with sufficient accuracy, can be used to assist in the identification of the corresponding geological structures.
Accordingly, although for most purposes the gravitational acceleration at the surface of the earth appears relatively constant, the gravitational acceleration does in fact change from place to place. These changes in the gravitational acceleration result from variations in the density of the material of the earth (or other celestial bodies such as an asteroid, moon or the like). For example, a measurement of the gravitational acceleration taken above a network of large underground caves (i.e., areas of relatively little mass) will be less than a similar measurement taken above a large dense deposit of nickel.
Compared to the Earth's gravitational acceleration at the surface, the variations in the gravitational acceleration are quite small. The nominal gravitational acceleration at the earth's surface resulting primarily from the Earth's mass is 9.81 meters per second per second (m/s2). A unit often used for acceleration is 1 g, which is defined as 9.81 m/s2. Variations in the gravitational acceleration are often measured in milligals (mgals) which is defined as 10−5 m/s2, and which is approximately equal to one-millionth of 1 g. The gravity gradient, which is defined as the rate of change of gravitational acceleration with respect to distance, has the units of m/s2/m (or 1/s2). For convenience, a defined unit for gravity gradient is the Eotvos, where one Eotvos (Eö) is defined to be equal to 10−9 m/s2/m, i.e., 10−9/s2, or 10−4 mgals/meter.
The gravitational acceleration due to an object decreases as the inverse of the square of the distance from the object (i.e., as the distance from the object doubles, the gravitational acceleration due to the object decreases by a factor of four) and it increases in direct proportion to the mass of the object. The direction of the gravitational acceleration depends on the distribution of mass within the object. Far from the object the gravitational acceleration is directed towards the center of mass of the object. However, close to the surface of the object, the strength and direction of the gravitational acceleration depends on the detailed distribution of the mass variation near the surface of the object. For example, near the surface of the earth, the gravitational acceleration due to the Earth will vary according to the mass distribution near the surface. Near, for example, the bottom of a mountain, the gravitational acceleration will have a small component directed horizontally towards the mountain, as well as the larger component directed vertically towards the center of the Earth. Near the top of the mountain the horizontal component would be greatly reduced. In three-dimensional space, the gravitational acceleration can be expressed as a vector having three elements (one for each direction): gx, gy and gz. The magnitude of these three components, and hence the magnitude and direction of the overall gravitational acceleration will thus vary spatially according to the detailed distribution of the mass within the gravitating object. The gravity gradient (G) is a measure of the rate of change of the gravitational acceleration with respect to distance. So, for example, as the gravitational acceleration is measured at different locations, the values of the components gx, gy, and gz will vary. In general each of these three gravitational acceleration components will vary with each of the three spatial coordinates. This leads to a nine component tensor for the gravity gradient, G. The components of the gravity gradient tensor are distinguished symbolically according to which gravitational acceleration component is being considered and which spatial direction is being considered. Thus the symbol Gxz is used to identify the rate of change of the gx acceleration component with changes in the vertical direction (z). As an example, if the gravitational acceleration component, gx, is measured along a line running vertically from the bottom of a valley, and the mountain narrows from its base to its peak, the gravitational field measured in the x-direction will decrease. The rate of change of gx with vertical distance is represented by Gxz. The gravitational component in the x-direction will also change along a horizontal line (i.e., constant z) becoming larger closer to a mountain. This gravity gradient component is represented by the symbol Gxx. Finally, the gravitational acceleration in the x-direction will in general vary along a horizontal line in the y-direction. This gravity gradient component will be represented by Gxy. Similarly, changes in the gravitational acceleration for the remaining two components (gy and gz) as measurements are taken in each of the three directions will be represented by Gyx, Gyy, Gyz, Gzx, Gzy and Gzz, respectively. The combination of these nine gravity gradient components form what is known as the gravity gradient tensor.
Gzz, the vertical gravity gradient at the Earth's surface, is about 3000 Eö, i.e., 3×10−6 m/s2/m, whereas perturbations in Gzz due to mineral deposits can be in the range of 1 Eö to 100 Eö.
It is a fundamental law of physics that, at an infinitesimal point, acceleration due to gravity is indistinguishable from acceleration due to other causes. That is, any device capable of detecting the acceleration due to gravity will also respond to acceleration due to other causes. Because of this, currently available devices capable of sensing acceleration with sufficient resolution and accuracy to detect the variations in the gravitational acceleration due to geological structures are typically land-based stationary instruments, as opposed to instruments mounted in a moving craft or vehicle. This is required because of the above described difficulty in distinguishing variations in gravitational acceleration caused by geological structures from the acceleration of the moving craft or vehicle in which the device is carried.
A gravimeter is a device sometimes used in geological surveying to measure the Earth's gravitational acceleration. By repeating the measurements at many locations a map of the gravitational acceleration can be obtained, which can then be used to locate geological features. A simple gravimeter is essentially an accelerometer (a device for measuring accelerations) such as a mass supported on a spring and constrained to move in only one direction, e.g., aligned in the vertical or z-direction along the axis of the spring. An acceleration along this z-axis causes the spring to deflect. The deflection can be detected to produce an output proportional to the acceleration in the z-direction less the acceleration due to gravity in the same axis (i.e., an output of az−gz).
As described above, the variation in the gravitational acceleration due to an anomaly is very small in comparison to the background gravitational acceleration and also often small in comparison to the acceleration of the vehicle. Since a gravimeter cannot distinguish between accelerations of a moving vehicle and changes in gravitational acceleration (which can be several orders of magnitude smaller), accurate measurements of the variation in the gravitational acceleration through use of an instrument in a mobile vehicle is extremely difficult. Attempts to remove the vehicle component of the measured acceleration, (e.g., through use of the global positioning system (GPS), to generate an approximation of the accelerations of the vehicle), have produced improvements but have not led to systems with a high enough resolution for effective airborne exploration, particularly for mineral deposits.
It is well recognized that an alternative to directly measuring gravitational acceleration from a mobile vehicle is to directly measure one or more components of the gravity gradient tensor, referenced above. Measuring the gravity gradient components can have considerable advantages.
It has been noted that while variations in the gravitational acceleration caused by a density anomaly may be small in comparison to the background gravitational acceleration, the relative perturbation in the gravity gradient created by a density anomaly near the surface relative to typical gravity gradients at the Earth's surface can be much larger. The local gravitational acceleration (which depends on the mass of an object and the proximity to that mass) falls off with the square of the distance to that mass (Newton's law of gravitation), whereas the gravity gradient (which is a spatial derivative) falls off with the cube of the distance from the mass. As a result, it has been shown that measuring the gravity gradient directly has advantages for locating geological features that lie within a few kilometers of the Earth's surface.
Referencing FIG. 9, a simple gravity gradiometer 1300 (a device for measuring a gravity gradient) is a balance beam 1302 with equal masses on either side of a pivot point 1304 and a torsion spring resisting rotation. If there is no gravity gradient (i.e., the gravitational acceleration is uniform), the gravitational forces on the masses would be equal on both sides of the pivot point and there would be no rotation of the beam. However, in a non-uniform gravity field, the balance beam, if not vertical, will rotate about the pivot 1304 with the one side of the beam being influenced by a stronger gravitational force m(g0+Δg) and the other side of the beam being influenced by a relatively lesser gravitational force m(g0). The amount of deflection (which is likely very small) is proportional to the difference (i.e., to the gravity gradient multiplied by the moment arm), and is inversely proportional to the rotational stiffness of the pivot. A translational acceleration of the pivot, and hence the balance beam, will cause no rotation. Therein lies a principal advantage of the gravity gradiometer.
An important improvement on the single beam gravity gradiometer is the two-beam “crossed dumbbell” gravity gradiometer. In such a gravity gradiometer, the dumbbells could be simple rectangular bars (FIG. 6).
Under the influence of the nominal vertical gravity gradient Gzz near the Earth's surface, the dumbbells will scissor (i.e., rotate in opposite directions) to an equilibrium position. If the instrument is moved to a location above an excess mass causing a greater Gzz, the bars will close slightly to a new equilibrium position.
However, almost all gravity gradiometers, including the dumbbell type of gravity gradiometer, when mounted in a moving vehicle will experience some disturbances as a result of displacement of the vehicle from a desired path and internal vibrations of the components of the vehicle. These disturbances can cause the sensor components to vibrate, generating random and potentially large rotations of the beams, making it difficult to resolve the beam rotations due to the gravity gradient.
A system that additionally addresses the problems of vehicle displacement from its ideal path and vibrations as noted above is desired. Current analysis indicates that such a system will provide improved measurement of the gravity gradients over current systems and will be of significant advantage in operation, particularly for geophysical exploration.